Abstract

A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic.

1. Introduction

We begin with finite and undirected simple graph of order . The set of vertices in a graph is called a dominating set if every vertex is either an element of or is adjacent to an element of . A dominating set is a minimal dominating set (MDS) if no proper subset is a dominating set.

The minimum cardinality of a dominating set of is called the domination number of which is denoted by and the corresponding dominating set is called a -set of .

The open neighborhood of is the set of vertices adjacent to and the closed neighborhood of is the set .

The complement of is the graph with vertex set and two vertices are adjacent in if and only if they are not adjacent in .

A subset is called a global dominating set in if is a dominating set of both and . The global domination number is the minimum cardinality of a global dominating set in . The concept of global domination in graph was introduced by Sampathkumar [1].

The upper bounds of global domination number are investigated by Brigham and Dutton [2] as well as by Poghosyan and Zverovich [3], while the global domination number of Boolean function graph is studied by Janakiraman et al. [4]. The global domination decision problems are NP-complete as discussed by Carrington [5] and by Carrington and Brigham [6]. The edge addition stable property in the context of global domination and connected global domination for cycle and path is discussed by Kavitha and David [7]. The concept of total global dominating set was introduced by Kulli and Janakiram [8] and they have also characterized total global dominating sets.

The wheel is defined to be the join . The vertex corresponding to is known as apex vertex and the vertices corresponding to cycle are known as rim vertices.

A shell graph is the graph obtained by taking concurrent chords in a cycle . The vertex at which all the chords are concurrent is called the apex. The shell graph is also called fan that is, .

Definition 1.1. The one-point union of cycles of length denoted by is the graph obtained by identifying one vertex of each cycle. The one-point union of cycles is known as friendship graph which is denoted by .

Definition 1.2 (see Shee and Ho [9]). Let be a graph and let , be copies of a graph . Then, the graph obtained by adding an edge from to for is called path union of .

Definition 1.3. Consider two copies of a graph , namely, and . Then, is the graph obtained by joining the vertices of degree of each graph to a new vertex .

Definition 1.4. Consider -copies of a graph , namely, . Then, is the graph obtained by joining the vertices of degree of each and to a new vertex , where .

Definition 1.5. Consider two copies of a graph , namely, and . Then, is the graph obtained by joining the vertices of degree of each graph by an edge as well as to a new vertex .

Definition 1.6. Consider -copies of a graph , namely, . Then, is the graph obtained by joining the vertices of degree of each and by an edge as well as to a new vertex , where . For the various graph theoretic notations and terminology we follow West [10], while the terms related with the concept of domination are used in the sense of Haynes et al. [11]. In the present paper we investigate some results on global dominating sets in the context of one-point union of cycles and path union of some graph families.

2. Main Results

Theorem 2.1. If is a one-point union of finite number of copies of cycle , then .

Proof. Consider -copies of cycle in and let be the vertex in common for these -copies. Because is adjacent to every other vertex in , is not adjacent to any other vertex in the complement of . Therefore, any dominating set for the complement of must contain . Hence, any global dominating set of must contain . Let and be the vertices of a cycle other than in . Now, since vertices both and are adjacent to third vertex of a cycle in , at least three vertices are essential to dominate which also dominate . Therefore, . Moreover, is the only -set of as is the only vertex of degree in . Hence, .Thus, as required.

Theorem 2.2. If is a one-point union of finite number of copies of cycle , then for .

Proof. Consider -copies of cycle in and let be the vertex in common for these -copies. Since is the vertex with maximum degree, it must belong to any global dominating set corresponding to for minimum cardinality. Now, it is easy to observe that for , . Therefore, any global dominating set corresponding to must contain vertices from each copy of other than the vertex for the minimum cardinality. Thus, for . For , we argue that any two nonadjacent vertices of dominate . If is a one-point union of - copies of , then the vertex , being a common vertex of , and a vertex nonadjacent to from each -copy of is the minimum vertices which dominate . These vertices also dominate because a vertex nonadjacent to from any two copies of are enough to dominate . Therefore, .Thus, for . The set, being a global dominating set with minimum cardinality, is an MDS with minimum cardinality. that is, the set is also a -set of . Thus, for as required.

Theorem 2.3. Let be a path union of finite number of copies of the graph of order having at least one vertex of degree . Then, is a -set of if and only if is a global dominating set of . Also, .

Proof. Let be the vertices of degree from each , respectively. Suppose that is a -set of . Now from the adjacency of the vertices in , it is easy to observe that is the only -set of except for the path union of . In case of , a set comprises of one vertex of degree from each or a vertex from each will be a -set. Because each vertex of dominates all the vertices of except its adjacent vertices and these adjacent vertices are dominated by or , consequently any two adjacent vertices are enough to dominate all the vertices of . Hence, is a dominating set of . Now, being a -set of , it is also a dominating set of . Thus, is a dominating set of both and , which implies that is a global dominating set of . Conversely, suppose that is a global dominating set of , that is, is a dominating set of both and .Consider . On removing any of the vertex from , the set will not dominate . Therefore, is not a dominating set of . This implies that is an MDS of . Each being a vertex of degree , clearly is of minimum cardinality. Hence, is a -set of . Thus, we have proved that is a -set of if and only if it is a global dominating set of . Since being a -set of , it is an MDS with minimum cardinality which implies that as required.

Theorem 2.4. Let . Then,(i) where can be , and (ii) where can be , , , .

Proof. Let and denote the two copies of the graph . Let and be the vertices of degree in and respectively. Suppose that is the vertex adjacent to both and .Consider . Let be either or . Then in order to dominate vertex in , either or is required. If , then and any other vertex from will dominate . Clearly, is a -set of . Hence, . For , is also a dominating set of which implies that is a global dominating set of and is of minimum cardinality. Consequently, . Thus, as required. Let be , , , or . Since and are the vertices of degree and they are adjacent to the vertex , is the -set of which implies that . being a -set and is adjacent to both and , is a global dominating set of which is of minimum cardinality. This implies that .Hence, as required.

Theorem 2.5. If with where can be , , , , or , then .

Proof. Let denote -copies of graph and let be the vertex of degree in . Suppose that are the vertices such that and are adjacent to where .Consider . Construct a dominating set as follows, Select a set of vertices of degree from each , respectively where dominates . Then, , is a dominating set of . If we remove any of from , then will not dominate which implies that is not a dominating set of . Consequently, is a minimal dominating set. Each being a vertex of degree , clearly is an MDS with minimum cardinality . Therefore, . We claim that is also a dominating set of because any two vertices and with are enough to dominate . Hence, is a dominating set of both and , which implies that .Thus, as required.

The following Theorem 2.6 can be proved by the arguments analogous to the above Theorem 2.5.

Theorem 2.6. If with where can be , , , , or , then .

On considering the two copies in Theorem 2.6, we have the following result.

Theorem 2.7. If where can be , , , , or , then .

Proof. Let and denote the two copies of the graph . Let and be the vertices of degree in and , respectively and let be the vertex adjacent to and both. Consider the graph . Then as and both being the vertices of degree in and respectively, the set is an MDS with minimum cardinality, that is, is a -set of which implies that . Now, is not a dominating set of because the vertex is adjacent to both and . But is a dominating set of which is also a dominating set of , that is, is a global dominating set of . To dominate the vertex in , either or is required in which is also in a global dominating set of . To dominate , if , then a vertex will be adjacent to both and any vertex from . Therefore, at least three vertices are required to dominate . Consequently, is a global dominating set of with minimum cardinality, that is, .Hence, as required.

3. Concluding Remarks

We have investigated some results corresponding to the concept of global domination. We have taken up the issue to obtain the global domination number for the larger graph obtained from the given graph. We have established the relations between the domination number and the global domination number for the graphs obtained by some graph operations, namely, one-point union and path union of graphs. More exploration is possible in the context of different graph families.

Acknowledgment

The authors are highly thankful to the anonymous referees for their critical comments and kind suggestions on the first draft of this paper.